UNIVERSITY OF NOTTINGHAM
Discussion Papers in Economics
________________________________________________
Discussion PaperNo. 06/11
Simple, Robust and Powerful Tests of the Breaking Trend
Hypothesis*
by David I. Harvey, Stephen J. Leybourne and A.M. Robert Taylor,
School of Economics, University of Nottingham
July 2006
_________________________________________________________
Simple, Robust and Powerful Tests of The
Breaking Trend Hypothesis
David I. Harvey, Stephen J. Leybourne and A. M. Robert Taylor
School of Economics, University of Nottingham
July 2006
Abstract
In this paper we develop a simple procedure which delivers tests for the pres-ence of a broken trend in a univariate time series which do not require knowledge of the form of serial correlation in the data and are robust as to whether the shocks are generated by an I(0) or an I(1) process. Two trend break models are considered: the …rst holds the level …xed while allowing the trend to break, while the latter allows for a simultaneous break in level and trend. For the known break date case our proposed tests are formed as a weighted average of the optimal tests appropriate for I(0) and I(1) shocks. The weighted statistics are shown to have standard normal limiting null distributions and to attain the Gaussian asymptotic local power envelope, in each case regardless of whether the shocks are I(0) or I(1). In the unknown break date case we adopt the method of Andrews (1993) and take a weighted average of the statistics formed as the supremum over all possible break dates, subject to a trimming parameter, in both the I(0) and I(1) environments. Monte Carlo evidence suggests that our tests are in most cases more powerful, often substantially so, than the robust broken trend tests of Sayginsoy and Vogelsang (2004). An empirical application highlights the practical usefulness of our proposed tests.
Keywords: Broken trend; power envelope; unit root; stationarity tests. JEL Classi…cation: C22.
We are grateful to Peter Phillips, Patrick Marsh and participants at the A.R. Bergstrom Memorial Conference held at the University of Essex, 24th and 25th May, 2006, for helpful comments on an earlier version of this paper. Correspondence to: Robert Taylor, School of Economics, University of Notting-ham, University Park, NottingNotting-ham, NG7 2RD, U.K. E-mail: [email protected]
1
Introduction
The focus of this paper is on testing for structural change in the trend function of a univariate time series. This is an important practical problem because the typical macroeconomic series appears to be characterised by temporary (I(0)) or permanent (I(1)) shocks ‡uctuating around a broken (segmented) trend: see, inter alia, Stock and Watson (1996,1999,2005) and Perron and Zhu (2005). It is clearly important to ade-quately model the trend function and failure to do so will lead to inconsistent estimates and poor forecasts. A further interesting application of testing for structural change in the trend function is discussed in Sayginsoy and Vogelsang (2004) [SV, hereafter], and concerns the important empirical debate as to whether convergence in per capita incomes among U.S. regions levelled o¤ in the mid-1970s, which can be explored by modelling the trend function in each region as having a slope shift in the mid-1970s; see SV for a number of key references in this literature. Segmented trends have also been fruitfully employed in the continuous time macroeconomic modelling literature by Nowan (1998), extending earlier work in Bergstrom et al. (1992).
Formal testing of whether a time series contains a broken trend function is greatly complicated by the fact that in practice it is not known whether the driving shocks are I(0) or I(1). If one knew that the shocks were I(0) then one could test for structural change in the trend function based on the level of the data. Similarly, if it were known that the shocks were I(1) then one could perform structural change tests on the …rst di¤erences of the data (growth rates). However, tests based on growth rates display very poor power properties relative to those based on levels (see Theorems 1 and 3 below) when the shocks are in fact I(0), as is discussed in a wider context in Vogelsang (1998). Moreover, as is shown later, the large sample null distributions of tests on the parameters of the trend function in levels data depend on whether the shocks are I(0) or I(1).
It is also well known that un-modelled trend breaks can bias unit root tests towards the non-rejection of the unit root hypothesis when the errors are I(1) (see, inter alia, Perron, 1989), while including unnecessary broken trends greatly reduces power to reject the unit root null under I(0) errors (see, for example, Marsh, 2005). Similarly, un-modelled trend breaks also cause spurious rejections in stationarity tests, such as that of Kwiatkowski et al. (1990)[KPSS, hereafter]. Where the potential trend break date is known, Perron (1989) shows that pivotal unit root inference can be achieved by including appropriate dummy variables in the relevant unit root regression. However, where the potential break date is unknown, as will usually be the case in practice, existing unit root tests which are based on search procedures, such as those of Zivot and Andrews (1992), are not similar, even asymptotically, (i.e. do not have pivotal limiting null distributions in the presence of trend breaks) with respect to the magnitude of the trend break, and often display poor power against I(0) shocks; see, inter alia, Nunes et al. (1997) and Vogelsang and Perron (1998). A circular testing problem therefore arises between tests on the parameters of the trend function and unit root/stationarity tests, as might also be expected in the light of the theoretical results in Phillips (1998).
In this paper we propose powerful and serial correlation robust tests for the presence of a structural break in the trend function of a univariate time series process. Our proposed tests do not require knowledge of the form of serial correlation in the data; in particular, no prior knowledge is needed as to whether the shocks are I(0) or I(1), thereby breaking the circular testing problem discussed above. Our test statistics are formed as a weighted average of the regression t-statistics for a broken trend appropriate for the case of I(0) and I(1) shocks; that is, a weighted average of the trend break t-statistics from a regression in levels and a regression in growth rates. The weighting function we employ is based on the KPSS stationarity statistics applied to the levels and growth rate data. In the known break date case the trend function and KPSS statistics are based around the true break date and the resulting weighted statistics have standard normal limiting null distributions and achieve the relevant Gaussian asymptotic local power envelope under both I(0) and I(1) shocks. Where the break date is unknown we follow Andrews (1993) and take the supremum of the trend function t-statistics, calculated for all possible break dates (subject to trimming at the ends of the sample), for both the I(0) and I(1) environments. In this case the KPSS statistics used in the weighting function are evaluated using an estimator of the breakpoint which is consistent regardless of whether the shocks are I(0) or I(1). A correction, of the form used in Vogelsang (1998), is required in the unknown break date case to ensure that, for a given signi…cance level, the weighted test has the same asymptotic critical value regardless of whether the shocks are I(0) or I(1). In both the known and unknown break date settings our proposed tests are made robust to short memory serial correlation in the shocks via the use of standard non-parametric long run [LR] variance estimators.
The remainder of the paper is organised as follows. Section 2 introduces our basic trend break model and outlines the assumptions underlying the model. Section 3 out-lines our proposed test statistics for a broken trend, both for the known and unknown break date cases, and establishes the large sample properties of these statistics. In section 4 we extend the reference model of section 2 to allow for the possibility of a simultaneous break in level and trend, and develop corresponding test statistics for this case. Practical issues relating to the computation of our proposed statistics, including tabulations of relevant critical values and scaling constants, are discussed in section 5. In section 6 we present an evaluation of the …nite sample size and power properties of our proposed tests, comparing these to the tests advocated in SV. Section 7 provides an empirical application to a variety of U.S. macroeconomic and …nancial data. Section 8 concludes. Proofs of our key results are gathered in a mathematical appendix.
In what follows we use the following notation: ‘x := y’(‘x =: y’) to indicate that x is de…ned by y (y is de…ned by x); b c to denote the integer part of the argument; ‘!’ and ‘p !’ denote convergence in probability and weak convergence, respectively,d as the sample size diverges to positive in…nity; I( ) to denote the indicator function, and N (a; b) to denote a Gaussian distribution with mean a and variance b. Finally, reference to a variable being Op(Tk) is taken to hold in its strict sense, meaning that
2
The Trend Break Model
Initially we consider the following trend break data generation process (DGP), referred to as “Model A” in what follows:
yt = + t + DTt( ) + ut; t = 1; :::; T; (1)
ut = ut 1+ "t; t = 2; :::; T; u1 = "1 (2)
where DTt( ) := I(t > T )(t T ), with T := b T c the (potential) trend break date
with associated break fraction 2 (0; 1). The error process "tin (2) is taken to be an
I(0)process, satisfying Assumption 1 below. An extended version of this model which allows for a simultaneous break in level will be considered subsequently in section 4. In (1), a break in trend occurs at time T when 6= 0. The I(0) scenario for ut is
represented by j j < 1 in (2), while the I(1) scenario obtains for = 1. Our interest in this paper therefore centres on testing the null hypothesis H0 : = 0 against the
two-sided alternative hypothesis1 H
1 : 6= 0, without assuming knowledge of whether
ut in (1) is I(0) or I(1).
We assume in what follows that "tin (2), satis…es Assumption 1 of SV (2005,pp.2-3);
that is,
Assumption 1. The stochastic process f"tg is such that
"t= c(L) t; c(L) = 1 X i=0 ciLi with c(1)2 > 0 and P1
i=0ijcij < 1, and where f tg is a martingale di¤erence sequence
with unit conditional variance and suptE( 4t) < 1.
Remark 1. Under the conditions of Assumption 1, the LR variance of "t is given by
!2
" := limT !1T 1E(PTt=1"t)
2 = c(1)2. Moreover, in the I(0) case the LR variance of
ut is given by !2u := limT !1T 1E(
PT
t=1ut)2 = !2"=(1 )2. Both these LR variances
play important roles in our subsequent analysis.
3
Tests for a Break in Trend
3.1
Known Break Fraction
In this section we consider …rst the case where the true break fraction, , is known. The case where the break fraction is unknown will be subsequently discussed in section 3.2. Under a known break fraction, we may partition H1 into two scaled components:
1One-sided hypotheses can also be accommodated within our framework. However, we shall not
discuss such tests further as it seems unlikely that the direction of any trend break would be known to the practitioner, a priori, particularly in the case of an unknown break date.
H1;0 : = T 3=2 when ut is I(0), and H1;1 : = T 1=2 when ut is I(1), where in
each case is a …nite non-negative constant. As we shall see below, these provide the appropriate Pitman drifts on under I(0) and I(1) errors, respectively. Notice that both H1;0 and H1;1 reduce to H0 when = 0.
Consider …rst the case where ut in (2) is known to be I(0) with = 0 in (2) and
"t a Gaussian white noise. Here the optimal (uniformly most powerful unbiased) test
of H0 against H1 rejects for large values of the absolute value of the t-ratio associated
with when (1) is estimated using OLS. That is, jt0( )j where2
t0( ) := ^( ) r ^2( )hfPTt=1xDT;t( )xDT;t( )0g 1 i 33 ; (3) ^( ) := 2 4 ( T X t=1 xDT;t( )xDT;t( )0 ) 1 T X t=1 xDT;t( )yt 3 5 3 with xDT;t( ) := f1; t; DTt( )g0, ^2( ) := T 1 PT t=1u^t( )2 and ^ut( ) := yt ^ ^ t ^( )DTt( ).
Correspondingly, if ut is known to be I(1), so that = 1 in (2), and ut is a
Gaussian white noise process, then the optimal test is based on the absolute value of the t-ratio associated with when (1) is estimated via OLS in …rst di¤erenced form. That is, writing
yt = + DUt( ) + ut; t = 2; :::; T (4)
where DUt( ) := I(t > T1), the optimal test rejects for large values of jt1( )j, where
t1( ) := ~( ) q ~2( )[fPTt=2xDU;t( )xDU;t( )0g 1]22 ; (5) ~( ) := 2 4 ( T X t=2 xDU;t( )xDU;t( )0 ) 1 T X t=2 xDU;t( ) yt 3 5 2
with xDU;t( ) := f1; DUt( )g0, ~2( ) := (T 1) 1PTt=2v~t( )2, and ~vt( ) := yt
~ ~( )DU
t( ).
In order to deal with more general I(0) and I(1) processes for ut, as are allowed
under Assumption 1, we need to replace ^2( ) and ~2( ) in the de…nitions of t0( )
of (3) and t1( ) of (5) with corresponding non-parametric LR variance estimators,
2The notation [:]
jj ([:]j) is used to denote the jj’th (j’th) element of the matrix (vector) within
^
!2( ) and ~!2( ), respectively, which are given by
^ !2( ) := ^0( ) + 2 T 1 X j=1 h(j=l)^j( ), ^j( ) := T 1 T X t=j+1 ^ ut( )^ut j( ) (6) ~ !2( ) := ~0( ) + 2 T 2 X j=1 h(j=l)~j( ), ~j( ) := (T 1) 1 T X t=j+2 ~ vt( )~vt j( ):(7)
In the context of (6) and (7), h( ) is a kernel function with associated bandwidth parameter `. In what follows we shall make use of the Bartlett kernel for h( ), such that h(j=`) := 1 j=(` + 1), with bandwidth parameter ` = O(T1=4).3 In the sequel, unless otherwise stated, any reference to t0( ) or t1( ) will be taken to imply those
based on the LR variance estimators in (6) and (7). Other choices of the kernel and bandwidth parameter could also be used, however, provided they satisfy standard regularity conditions, such as are outlined in Assumptions A3 and either A4 or A4’of Jansson (2002,pp.1450,1452), respectively.
The following Theorem establishes the asymptotic behaviour of the jt0( )j and
jt1( )j statistics under both H1;0 and H0;1.
Theorem 1 Let the time series process fytg be generated according to (1) and (2),
and let Assumption 1 hold.
(i) If ut in (2) is I(0) (i.e. j j < 1), then: (a) jt0( )j d ! jL00( ; )j, where L00( ; ) := f R1 0 RT (r; ) 2dr g1=2 !u + R1 0 RT (r; )dW (r)dr fR01RT (r; ) 2drg1=2 ; and (b) jt1( )j = Opf(l=T )1=2g.
(ii) If ut in (2) is I(1) (i.e. = 1), then: (a) jt0( )j = Opf(T=l)1=2g, and (b)
jt1( )j d ! jL11( ; )j where L11( ; ) := f R1 0 RU (r; ) 2dr g1=2 !" + R1 0 RU (r; )dW (r)dr fR01RU (r; )2drg1=2
where W (r) is a standard Brownian motion on [0; 1], and RT (r; ) is the continuous-time residual from the projection of (r )I(r > ) onto the space spanned by f1; rg, and RU (r; ) is the residual from the projection of I(r > ) onto f1g.
Remark 2. It is trivially seen from the results in Theorem 1 that under H0 : = 0,
t0( ) d ! N(0; 1) if ut is I(0), while t1( ) d ! N(0; 1) if ut is I(1). Consequently, 3Notice that ^ 0( ) = ^ 2 ( ) and ~0( ) = ~2( ).
with knowledge of the order of integration of ut, the appropriate two-sided test can be
implemented using critical values from the standard normal distribution.
Remark 3. From the results in part (i) of Theorem 1 it is seen that when ut is I(0)
jt1( )j converges in probability to zero, regardless of the value of , while jt0( )j
at-tains the Gaussian asymptotic local power envelope for this testing problem. Similarly, from the results in part (ii) of Theorem 1 it is seen that when utis I(1), jt1( )j achieves
the I(1) Gaussian asymptotic local power envelope, while jt0( )j diverges irrespective
of the value of .
In view of the above results, and given that the order of integration of ut is not
known in practice, it is a fairly natural step to consider constructing a procedure that employs some auxiliary routine which ensures that, asymptotically at least, the statistic jt0( )j of (3) is selected when utis I(0) while jt1( )j of (5) is selected when ut is I(1),
thereby ensuring that the asymptotically optimal test is selected in the limit. To that end we pursue an approach based on a data-dependent weighted average of jt0( )j and
jt1( )j of the form
t :=f (S0( ); S1( )) jt0( )jg + f[1 (S0( ); S1( ))] jt1( )jg (8)
In (8), S0( ) and S1( ) are auxiliary statistics chosen such that, as the sample size
diverges to positive in…nity, the weight function ( ; ) converges to unity when ut is
I(0) and to zero when ut is I(1), such that t will collapse to jt0( )j when ut is I(0),
and to jt1( )j when ut is I(1). Because the auxiliary routine needs to be ambivalent
between H0and H1, the S0( )and S1( )statistics must also be invariant with respect
to , and in (1).
We therefore need to chose appropriate auxiliary statistics, S0( ) and S1( ), and
weight function, ( ; ). For the former we shall adopt the stationarity statistics of KPSS calculated from the residuals f^ut( )gTt=1 and f~vt( )gTt=2, respectively, each of
which are exact invariant to , and . Speci…cally,
S0( ) := PT t=1 Pt i=1u^i( ) 2 T2!^2( ) ; S1( ) := PT t=2 Pt i=2v~t( ) 2 (T 1)2!~2( ) (9)
where ^!2( ) and ~!2( ) as as de…ned in (6) and (7) respectively. The relevant large sample properties of these two statistics are given in the following Lemma.
Lemma 1 Let the conditions of Theorem 1 hold.
(i) If ut is I(0) then: (a) S0( ) = Op(1), and (b) S1( ) = Op(`=T ).
(ii) If ut is I(1) then: (a) S0( ) = Op(T =`), and (b) S1( ) = Op(1).
The results in Lemma 1 therefore suggest a weight function, ( ; ), of the form (S0( ); S1( )) := exp[ fg1S0( )S1( )gg2] (10)
where g1 and g2 are positive constants, since this will clearly converge to unity when ut
is I(0) and to zero when utis I(1), as required. Moreover, it does so at an exponential
rate. Using the large sample results in Theorem 1 and Lemma 1, we are in a position to state the following Corollary.
Corollary 1 Let the conditions of Theorem 1 hold. (i) If utis I(0), then (S0( ); S1( ))
p
! 1 under both H0 and H1;0, and t =jt0( )j+
op(1) d
! jL00( ; )j.
(ii) If ut is I(1), then (S0( ); S1( )) p
! 0 under H0 and H1;1 and t = jt1( )j +
op(1) d
! jL11( ; )j.
Remark 4. The results in Corollary 1 show that if ut is I(0), t is asymptotically
equivalent to jt0( )j, while if ut is I(1), t is asymptotically equivalent to jt1( )j.
Consequently, t achieves the appropriate Gaussian asymptotic local power envelope regardless of whether utis I(0) or I(1). Moreover, under H0, t
d
! jN(0; 1)j irrespective of whether ut is I(0) or I(1), so that a two-sided test can again be implemented using
critical values from the standard normal distribution.
Remark 5. Notice from part (ii) of Corollary 1 that the product (S0( ); S1( ))
jt0( )j is of op(1) even though, as shown in part (ii) of Theorem 1, jt0( )j
di-verges at rate Opf(T=`)1=2g. This result is due to our choice of weighting function
(S0( ); S1( ))of (10) which converges in probability to zero at an exponential rate
in T when ut is I(1).
3.2
Unknown Break Fraction
We now consider the case where the true break fraction cannot be considered known, a priori. In this case we follow the approach of Andrews (1993) and consider statistics based on the maxima of the sequences of statistics4 fjt0( )j ; 2 g and fjt1( )j ; 2
g, where = [ L; U], with 0 < L < U < 1, where the quantities L and U will
be referred to as the trimming parameters, and where it is assumed throughout that 2 . De…ning :=fb LTc; :::; b UTcg, these statistics are given by
t0 := sup s2 jt 0(s=T )j (11) and t1 := sup s2 jt 1(s=T )j ; (12)
4Although we analyse tests based on the maxima of these sequences of statistics, tests based on the
corresponding mean- or mean-exponential-type statistics of, inter alia, Hansen (1992) and Andrews and Ploberger (1994), respectively, could also be used.
with associated breakpoint estimators of given by ^ := arg sups2 jt0(s=T )j and
~ := arg sups2 jt1(s=T )j, respectively, such that t0 jt0(^)j and t1 jt1(~)j. The
analogue of our t statistic of (8) is then given by
t :=f (S0(^); S1(^)) t0g + m f[1 (S0(^); S1(^))] t1g (13)
where m is a positive …nite constant whose precise role is discussed below. Observe that both stationarity statistics are evaluated at the breakpoint estimator ^, this being a consistent estimator of regardless of whether ut is I(0) or I(1).5
In the current context where the break fraction is unknown, it cannot be consis-tently estimated under the Pitman drift alternatives of the form considered in section 3.1. However, for the purposes of empirical work a rejection against a broken trend is clearly of rather limited use without a consistent estimate of where the break occurs. Consequently, we shall consider only …xed alternatives in this situation, where consis-tent estimation of the unknown break fraction is possible, establishing the consistency properties of our tests. However, in the case where ut is I(1) the test which rejects for
large values of t1 has an equivalent critical region to the likelihood ratio-type test in a linear setting of Andrews (1993) and, as such, will possess the weak local optimality property of Andrews (1993, Equation (5.5)). This need not be true for a test based on t0 because of the presence of trending regressors.
We …rst establish the large sample behaviour of the t0 and t1 statistics under the null hypothesis, H0 : = 0, when the shocks, ut, are either I(0) or I(1).
Theorem 2 Let the time series process fytg be generated according to (1) and (2)
under H0 : = 0, and let Assumption 1 hold.
(i) If ut is I(0), then: (a) t0 d
! sup 2 jL00( ; 0)j, and (b) t1 = Opf(`=T )1=2g.
(ii) If ut is I(1), then: (a) t0 = Opf(T=`)1=2g, and (b) t1 d
! sup 2 jL11( ; 0)j :
We now establish the consistency rates of these statistics under a …xed alternative of the form H1 : 6= 0.
Theorem 3 Let the time series process fytg be generated according to (1) and (2)
under H1 : 6= 0, and let Assumption 1 hold.
(i) If ut is I(0), then: (a) t0 = Op(T3=2), and (b) t1 = Opf(`T )1=2g.
(ii) If ut is I(1), then: (a) t0 = Op(T =`1=2), and (b) t1 = Op(T1=2).
Remark 6. It is interesting to note from the results in part (ii) of Theorem 3 that t0 diverges at a faster rate than t1when ut is I(1), which may seem counterintuitive given
that t1 would be thought of as the preferred test in this situation. However, it must
5An alternative, which makes no di¤erence to the large sample results which follow, is to use the
be borne in mind from part (ii) of Theorem 2 that t0 also diverges under H0 when ut
is I(1) while t1 has a well-de…ned critical region.
In order to derive the asymptotic behaviour of the weighted statistic t of (13) we must next establish the large sample behaviour of the S0(^) and S1(^) statistics. This
is done in the following lemma, the proof of which is straightforward but tedious given results established in Lemma 1 and Theorems 2 and 3 and is therefore omitted in the interests of brevity.
Lemma 2 Let the conditions of Theorem 1 hold.
(i). If ut is I(0), then: (a) S0(^) = Op(1), and (b) S1(^) = Op(`=T ).
(ii). If ut is I(1), then: (a) S0(^) = Op(T =`), and (b) S1(^) = Op(1).
An immediate corollary of the results in Lemma 2 is that, regardless of whether H0
or H1 holds, when utis I(0), (S0(^); S1(^)) p
! 1, while if utis I(1), (S0(^); S1(^)) p
! 0. Consequently, using the results in Theorems 2 and 3, we may state the following corollary concerning the large sample behaviour of our weighted statistic, t of (13), which again exploits the fact that convergence in probability of (S0(^); S1(^)), either
to unity or zero, occurs at an exponential rate. Corollary 2 Let the conditions of Theorem 1 hold.
(i) Let H0 : = 0 hold. Then: (a) if ut is I(0), t = t0 + op(1) d
! sup 2 jL00( ; 0)j;
(b) if ut is I(1), t = m t1+ op(1) d
! m sup 2 jL11( ; 0)j.
(ii) Let H1 : 6= 0 hold. Then: (a) if ut is I(0), t = t0+ op(1) = Op(T3=2); (b) if ut
is I(1), t = m t1+ op(1) = Op(T1=2).
It is seen from the results in part (i) of Corollary 2 that, in contrast to the known breakpoint case considered in section 3.1, the asymptotic null distribution of the weighted statistic t of (13) di¤ers as to whether utis I(0) or I(1). Moreover, in neither
case is this distribution standard normal. Similarly to Vogelsang (1998), however, we can choose the constant m in (13) such that, for a given signi…cance level under H0,
the critical value of m sup 2 jL11( ; 0)j coincides with that of sup 2 jL00( ; 0)j. This
then ensures that, for the chosen signi…cance level, the asymptotic null critical value of t is the same irrespective of whether ut is I(0) or I(1). Under H1 : 6= 0, it is
seen from Corollary 2 that t is consistent at rate Op(T3=2)when ut is I(0) and at rate
Op(T1=2) when ut is I(1).
4
Allowing for a Simultaneous Break in Level
Although trend breaks are the central concern of this paper, we might also consider extending our analysis to allow (but not test for) a break in level occurring at the same time as the break in trend. To this end, consider replacing (1) with
whose di¤erenced form, corresponding to (4), is given by
yt = + Dt( ) + DUt( ) + ut; t = 2; :::; T; (15)
where Dt( ) := I(t = T ). The shocks, ut, are still assumed to be generated according
to (2). In what follows we will refer to (14) and (2) together as “Model B”.
In section 4.1 we will initially consider the known break date case, with the unknown break date case subsequently discussed in section 4.2. In order to avoid unnecessarily complex notation, we will repeat the notation of section 3 for the quantities involved.
4.1
Known Break Fraction
For the known break fraction case we need place no restrictions on the value of under each of H0, H1;0 and H1;1, these being de…ned exactly as in section 3.1. We
consequently re-de…ne t0( ) as follows:
t0( ) := ^( ) r ^ !2( )hfPTt=1xDT;t( )xDT;t( )0g 1 i 44 ; ^( ) := 2 4 ( T X t=1 xDT;t( )xDT;t( )0 ) 1 T X t=1 xDT;t( )yt 3 5 4
with xDT;t( ) :=f1; t; DUt( ); DTt( )g0and ^!2( )calculated as in (6) but using the
OLS residuals ^ut( ) := yt ^ ^ t ^DUt( ) ^( )DTt( )from (14). Similarly,
t1( ) is re-de…ned to be t1( ) := ~( ) r ~ !2( )hfPTt=2xDU;t( )xDU;t( )0g 1 i 33 ; ~( ) := 2 4 ( T X t=2 xDU;t( )xDU;t( )0 ) 1 T X t=2 xDU;t( ) yt 3 5 3
with xDU;t( ) := f1; Dt( ); DUt( )g0 and ~!2( ) calculated as in (7) but using the
OLS residuals ~vt( ) := yt ~ ~Dt( ) ~( )DUt( )from (15).
In Theorem 4 we now establish the asymptotic behaviour of jt0( )j and jt1( )j
under both H1;0 and H1;1. The proof of Theorem 4 is a straightforward generalization
of that of Theorem 1 and is therefore omitted.
Theorem 4 Let the time series process fytg be generated according to (14) and (2),
(i) If ut is I(0), then: (a) jt0( )j d ! jLU;00( ; )j where LU;00( ; ) := f R1 0 RTU(r; ) 2dr g1=2 !u + R1 0 RTU(r; )dW (r)dr fR01RTU(r; )2drg1=2 ; and (b) jt1( )j = Opf(`=T )1=2g.
(ii) If ut is I(1), then: (a) jt0( )j = Opf(T=`)1=2g, and (b) jt1( )j d ! jL11( ; )j where L11( ; ) := f R1 0 RU (r; ) 2dr g1=2 !" + R1 0 RU (r; )dW (r)dr fR01RU (r; )2drg1=2
where RTU(r; ) is a continuous-time residual from the projection of (r )1(r > )
onto the space spanned by f1; r; 1(r > )g, and W (r) and RU(r; ) are as de…ned in Theorem 1.
Remark 7. As with the results in Theorem 1 for Model A, it is trivially seen that LU;00( ; )follows a Gaussian distribution, reducing to a standard normal distribution
under H0 and attaining the Gaussian asymptotic local power envelope under H0;1.
Remark 8. Observe from the result given in part (ii)(b) of Theorem 4 that the limiting distribution of jt1( )j from Model B is identical to that for Model A given in Theorem
1 (ii)(b). This is because the regressor Dt( )has an asymptotically negligible e¤ect on
jt1( )j. Consequently, the comments made in Remarks 2 and 3 relating to the jt1( )j
statistic in the context of Model A when ut is I(1) also apply under Model B.
In order to extend our t statistic of (8) to the case of a simultaneous break in level, we re-de…ne S0( ) and S1( ), (S0( ); S1( )), and t to be constructed as in
(9), (10) and (8), respectively, but constructed using the re-de…ned OLS residuals from (14) and (15). It is entirely straightforward to demonstrate that the orders given in Lemma 1 for S0( ) and S1( ) remain appropriate in the case of a simultaneous level
break. We may therefore state the following corollary. Corollary 3 Let the conditions of Theorem 4 hold.
(i) If ut is I(0), then under both H0 and H1;0, t = jt0( )j + op(1) d
! jLU;00( ; )j.
(ii) If ut is I(1), then under both H0 and H1;1, t = jt1( )j + op(1) d
! jL11( ; )j.
Remark 9. As with the results for the break in trend only case, t achieves the appropriate Gaussian asymptotic local power envelope regardless of whether ut is I(0)
or I(1). Moreover, we again have the result that t ! jN(0; 1)j under Hd 0, irrespective
4.2
Unknown Break Fraction
We now consider the case where is unknown in Model B. Here we proceed as in section 3.1, appropriately re-de…ning the various statistics involved to be formed from the OLS residuals from either (14) or (15), as appropriate.
As in SV (2004), the null hypothesis H0 must be re-stated as H0 : = = 0, in
the current context in order to obtain a pivotal limiting null distribution for our test statistic. The following theorem, whose proof is entirely similar to that of Theorem 2 and, hence, is omitted, details the large sample behaviour of the re-de…ned t0 and t1 statistics under H0.
Theorem 5 Let the time series process fytg be generated according to (14) and (2)
under H0 : = = 0, and let Assumption 1 hold.
(i) If ut is I(0), then: (a) t0 d
! sup 2 jLU;00( ; 0)j, and (b) t1 = Opf(`=T )1=2g.
(ii) If ut is I(1), then: (a) t0 = Opf(T=`)1=2g, and (b) t1 d
! sup 2 jL11( ; 0)j .
For …xed alternatives of the form H1 : 6= 0; with now unrestricted, it can be
shown that the rates of divergence given in Theorem 3 for t0 and t1 remain appro-priate here also, as do the rates of divergence of S0(^) and S1(~) given in Lemma 2.
Consequently, for t of (13) we …nd that under H0, when ut is I(0)
t = t0+ op(1) d ! sup 2 jL U;00( ; 0)j while if ut is I(1) t = m t1+ op(1) d ! m sup 2 jL 11( ; 0)j :
Under H1, when ut is I(0), we obtain that t is consistent at rate Op(T3=2), while if
ut is I(1), t is consistent at rate Op(T1=2). Again we must choose the constant m
such that, for a signi…cance level under H0, the critical value of m sup 2 jL11( ; 0)j
coincides with that of sup 2 jLU;00( ; 0)j. As before this ensures that the asymptotic
null critical values of t are the same regardless of whether ut is I(0) or I(1).
5
Practical Implementation of the Test Procedures
Asymptotic critical values for our proposed t tests for both Models A and B are provided in Table 1, along with the corresponding values of m . The values reported are for tests of the null of no break in trend against a two-sided alternative, with the potential break date unknown; that is, for testing H0 : = 0 against H1 : 6= 0 in the
context of Model A, and H0 : = = 0 against H1 : 6= 0 in the context of Model
B. As in SV, 10% trimming was used, such that L = 0:1 and U = 0:9. The results
approximations for T = 1000 and 50000 replications using the rndKMn random number generator of Gauss 6.0.
Table 1 about here
For the trend break tests to be operational, we also need to specify the constants g1 and g2 in (10). After Monte Carlo simulation of the …nite sample size and power of
the tests for a range of possible settings, we found that the choices g1 = 500, g2 = 2
and ` = b4(T=100)1=4
c gave the best overall performance, and we therefore recommend use of these values for practical application of the new procedure. These choices apply to both Model A and Model B.
6
Numerical Results
In this section we use Monte Carlo methods to investigate the …nite sample size and power performance of the t tests using Monte Carlo simulation. We focus on the unknown break date case - arguably the case of most practical interest - and again employ 10% trimming ( L= 0:1, U = 0:9) throughout. All the results reported in this
section are for two-sided tests conducted at the 0.05 nominal asymptotic signi…cance level, and were computed over 10000 replications, again using the rndKMn function of Gauss 6.0.
The data generating process [DGP] we use to conduct our simulations is a simpli…ed form of (14) and (2) given by:
yt = DUt( ) + DTt( ) + ut; t = 1; :::; T; (16)
(1 L)ut = (1 L)"t; t = 2; :::; T; u1 = "1 (17)
where "t IIN (0; 1). An investigation into the …nite sample size properties of our
proposed tests, relative to the tests advocated in SV, outlined immediately below, is provided …rst in section 6.1. The …nite sample power properties of the tests are subsequently explored in section 6.2, again relative to the tests of SV. Results are reported for both Model A and Model B, for samples of size T = 150 and T = 300. Our proposed t statistic is constructed as detailed in section 3.2 for the case of Model A, and as detailed in section 4.2 for the case of Model B. Recall that t tests the null hypotheses H0 : = 0 and H0 : = = 0 in the context of Models A and B,
respectively.
In an unpublished paper, SV also propose tests for a break in trend that are robust to strong serial correlation in the data. For a given possible break date Ta 2 ,
consider a standard Wald statistic of the null hypothesis H0 : = 0 for Model A, or
H0 : = = 0 for Model B, with the implicit long-run variance estimator constructed
using a Daniell kernel with bandwidth M = bbT c, b 2 (0; 1]. Denoting this Wald statistic by Wb(T
a), the superscript ‘b’ referring to the bandwidth fraction, the SV
approaches to overcoming the fact that the break date is unknown, i.e. one of the following quantities MeanWb := T 1 X Ta2 Wb(Ta) SupWb := sup Ta2 Wb(Ta):
To achieve robustness to the possibility of I(1) shocks, these quantities are then mul-tiplied by a correction factor based on a unit root statistic, following the approach of Vogelsang (1998). The unit root statistics considered are straightforward extensions of the variable addition statistic of Park (1990) and Park and Choi (1988), and the vari-ance ratio statistic of Breitung (2002), where the underlying regressions are augmented with dummy variables to model the possible break. Denoting these unit root statistics for a given break date, Ta 2 , by J (Ta) and BG(Ta) respectively, the SV statistics
are given by, where U R generically denotes either J or BG, MeanWU Rb := MeanWbexp U Rmean inf
Ta2
U R(Ta)
SupWU Rb := SupWbexp U Rsup inf
Ta2
U R(Ta) :
The constants BGmean, Jmean, BGsup and Jsup, which are speci…c to each test, are chosen such that for the given test and a given signi…cance level, , the critical values for the test coincide under I(0) and I(1) errors. For Model A, SV advocate using (i) MeanW0:02 BG
if ut is I(0), (ii) SupWBG0:10 if ut is I(1), while (iii) SupWBG0:06 is recommended where it
is not known if ut is I(0) or I(1), it giving the best overall power across both the I(0)
and I(1) cases. For Model B, the corresponding tests are (i) MeanW0:02
BG , (ii) SupWJ0:36,
and (iii) MeanW0:18
BG . Critical values and associated values of the BG mean, J mean, BG sup and J
sup constants are reported by SV for each of these tests.
6.1
Size Properties
Table 2 reports the empirical rejection frequencies (sizes), for the t test together with the recommended tests from SV for each of Models A and B. These were obtained by setting = = 0 in (16). The AR and M A noise parameters in (17) were varied over = 1 (c=T ) for c 2 f0; 10; 20; T g and 2 f0; 0:4; 0:8g, respectively. Notice that for c = T , ut is a pure M A(1) process.
Table 2 about here
In the case of I(1) shocks (c = 0) we see that the t test is somewhat oversized in …nite samples. This is a …nite sample e¤ect, as can be seen on comparing results for T = 150 and T = 300, suggesting that the asymptotic distributions can be poor approximations in relatively small samples. Conversely, in cases where the shocks are
I(0) (c > 0), the t test tends to be slightly under-sized (some exceptions are seen where = 0:8), particularly for c = T . The over-sizing e¤ect under I(1) shocks is in most cases (an exception occurs for = 0:8) little di¤erent between Models A and B, but the under-sizing seen in the I(0) case tends to be somewhat less pronounced under Model B. Interestingly, for a given value of c, the size of t does not appear particularly sensitive to the choice of in the case of Model A, although in the case of Model B there does appear to be some sensitivity in the case of = 0:8. The size properties of the SV tests are also sensitive to the choice of c and . For the case of I(1) shocks these tests can be seriously over-sized when = 0:8, but are better sized than t when 0. For I(0) shocks the pattern is mixed, depending on the value of and on the particular test involved, but in most cases a larger degree of under-size is seen in the SV tests than for t .
6.2
Power Properties
Figures 1–3 and 5–7 present results for the report the empirical rejection frequencies (powers) of the tests based on Models A and B respectively. The data were generated according to (16) for a grid of values, covering the range [0; 1] in steps of 0.02. Under Model A, = 0 in all experiments, while for Model B we set = 5 . We consider three break timings6
2 f0:25; 0:5; 0:75g, and focus on cases where there is most likely to be some ambiguity as to the order of integration by using = 1 (c=T )with c2 f0; 10; 20g in (17). In order to economise on space, we only report results for = 0. The results for 2 f 0:4; 0; 8g are qualitatively similar and are available on request.
Consider …rst Figures 1 and 5 which relate to the case of I(0) shocks (c = 0) in Models A and B respectively. Here we can see that t dominates the SV tests on power, enjoying a marked power advantage right across the range of the power function over the recommended tests of SV for both Model A and Model B and for all of the values of considered. For example, for T = 150 and = 0:25, while t e¤ectively has power of unity for = 1 under both Models A and B, none of the SV tests have power in excess of 25 % (15 %) under Model A (Model B). The power of the SV tests are also sensitive to the location of the trend break: speci…cally, their power is much lower, other things equal, for cases where the break is located away from the middle of the sample. In contrast, the power functions for t do not appear to depend to any noticeable degree on the location of the break. Comparing results between Models A and B we also see that, other things equal, the power functions of t are virtually identical for Models A and B, as predicted by the asymptotic theory; cf. Remark 8.
Figures 1 8 about here
Because the t tests are a little over-sized in the I(1) case (see the discussion in section 6.1), we also report size-adjusted powers for this case. The size-adjusted power curves are given in Figures 4 and 8 for Models A and B, respectively. The qualitative
conclusions drawn from Figures 4 and 8 are largely the same as those from Figures 1 and 5, with t clearly dominating all of SV’s advocated tests on power.
Consider next Figures 2 and 3 which graph, for Model A, the power curves of the various tests for c = 10 and c = 20, respectively. In each case, a comparison at the origin ( = 0) shows that all of the tests are under-sized, with the recommended tests of SV generally more so than t ; cf. Table 2. For both early and late break dates ( = 0:25and = 0:75) the t test again dominates all of the SV tests on power for both c = 10 and c = 20. Where a break occurs in the middle of the sample ( = 0:5) the t test again dominates the SV tests for T = 150, but for T = 300, it is seen that a small region occurs in the power curves where the SV tests display slightly higher power than t . For c = 10 and for SupWBG0:06(which is the appropriate test to compare
t with) this region is approximately from = 0:1 to = 0:2, while for c = 20 it is around = 0:1 to = 0:25. Much the same patterns are also seen under Model B in Figures 6 and 7, except that the SupWJ0:36 test is always dominated by t .
7
Empirical Application
We now consider applying the trend break tests to recent US macroeconomic time series data. Speci…cally, we examine quarterly seasonally adjusted real GDP observed for 1970Q1 to 2003Q4 (136 observations), and six monthly series observed for 1970M1 to 2003M12 (408 observations): seasonally adjusted unemployment rate, seasonally adjusted money supply M2, 3-month Treasury bill interest rate, commodities spot price index, seasonally adjusted consumer price index and average hourly earnings. The real GDP data was obtained from www.economagic.com, and the monthly series were taken from the database compiled by Stock and Watson (2005). All the variables are measured in logarithms and are plotted in Figure 9.
Table 3 and Figure 9 about here
Table 3 reports the results from application of the tests to these series at the = 0:01 and = 0:05 signi…cance levels. Consider …rst the tests applied using Model A. Our proposed t test fails to reject the null of no break in trend for real output and unemployment, but rejects at the 0.05-level for the other …ve series, of which the null hypothesis can be rejected at the 0.01-level for the money supply, consumer prices and earnings. Where rejections (at the 0.05-level or lower) are obtained, the estimated break dates suggested by the procedure are also reported in Table 3. These were obtained by taking a weighted average of the break date estimates ^ and ~, using the weight function (S0(^); S1(^)) employed in the t test, i.e. (S0(^); S1(^))^ +f1 (S0(^); S1(^))g~.
The estimated break dates are superimposed on the plots in Figure 9, and correspond well to break timings suggested by visual inspection. In contrast, none of the SV tests can reject the null at either signi…cance level for any of the series considered.
Turning now to Model B, the t results correspond very closely with those obtained using Model A. Rejections in favour of a break in trend at the 0.05-level are observed
for the money supply, interest rate, commodity prices, consumer prices and earnings, at the 0.01-level for the money supply, consumer prices and earnings, while the null is not rejected for real output and unemployment. Moreover, for the …ve series where a break is detected, the estimated break dates coincide with those found for Model A, except for consumer prices and earnings where they di¤er only by one observation. On the other hand, the SV tests again do not reject the null in general, especially when the more reliable MeanW0:18
BG test is used. Rejections are obtained at the 0.05-level for the
money supply and consumer prices series using the SupWJ0:36 test, but not elsewhere,
and it should be stressed that this test is recommended for practical use by SV only where it is known that the shocks are I(1).
8
Conclusions
In this paper we have proposed new tests for a broken trend, with or without a simulta-neous break in level, in a univariate time series process which do not require knowledge of the form of serial correlation in the data and are robust to whether the shocks are I(0)or I(1). Our proposed tests are based on simple data-dependent weighted averages of the absolute values of two conventional regression t-ratios, one appropriate for when the data are generated by an I(0) process and the other when the data are I(1). Under a known break date our proposed tests have standard normal limiting null distributions and achieve the relevant Gaussian power envelope, in both I(0) and I(1) environments. For the more realistic case of an unknown break date we employ a supremum-based approach as in, inter alia, Andrews (1993) and establish the large sample properties of the resulting tests which are shown to have pivotal null distributions and to deliver consistent tests, again regardless of whether the shocks are I(0) or I(1). Monte Carlo evidence was reported which suggested that our tests are in most cases more powerful, and often substantially so, than the recommended robust broken trend tests proposed in an unpublished paper by Sayginsoy and Vogelsang (2004). An empirical example to a variety of US macroeconomic data highlighted the practical usefulness of our pro-posed tests, uncovering signi…cant evidence of trend breaks in the majority of the series analysed.
To conclude we suggest three topics for possible further research. First, as dis-cussed in the introduction, extant unit root tests which allow for trend breaks have the undesirable property of not being similar, even asymptotically, with respect to the magnitude of the trend break. Moreover, they lose power if unnecessary trend break dummies are included in the vector of deterministic variables used to de-trend the data prior to computing the unit root statistic. It would therefore be interesting to conduct a formal analysis of the properties of unit root tests where the inclusion of trend break dummies or otherwise in the vector of deterministic variables was speci…ed on the basis of the outcomes of the tests considered in this paper, in particular to establish whether similar unit root tests can be obtained. Second, we have focused in this paper on the case of a single break in trend. It would also be useful to extend our analysis to
the case of multiple trend breaks. For the case of known break dates this would be a trivial extension of the results in this paper. An analysis of the unknown break dates case would be likely to be considerably more involved. However, we conjecture that it should be feasible using a sequential testing strategy, along the lines of that considered in Bai and Perron (1998,2003). Thirdly, we have restricted attention in this paper to the case where the shocks are either I(0) or I(1). An interesting extension of this paper would be to consider the more general scenario where the shocks are either I(0) (short memory) or fractionally integrated of order d, I(d), 0 < d 1 (long memory). In this case a bounds-type procedure, based on the results presented in this paper for the polar I(0) and I(1) cases, might be usefully explored.
Appendix
In what follows, due to invariance of the statistics concerned, we can set = = 0 without loss of generality.
Proof of Theorem 1.
(i)We …rst establish the result in (a). Using the Frisch-Waugh-Lovell Theorem (FWLT) we can write t0( ) in the form
t0( ) = + T 3=2PRT t( )ut T 3PRT t( )2 1 q ^ !2( )=T 3PRT t( )2
where RTt( ), t = 1; :::; T , are the OLS residuals from regressing DTt( )onto 1 and
t. Entirely standard results, including the fact that ^!2( )! !p 2u, allow us to establish the weak convergence result,
t0( ) d ! ( + !u R1 0 RT (r; )dW (r)dr R1 0 RT (r; )2dr ) 1 q !2 u= R1 0 RT (r; )2dr ; (A.1)
where W (r) is a standard Brownian motion, de…ned via, ! 1 u T 1=2
PbT rc
t=1 ut d
! W (r). The result in (a) is then established on rearranging (A.1).
Turning to the result in (b), and again using the FWLT, we can write t1( ) as
(T =`)1=2t1( ) = T 1=2+ P RUt( ) ut T 1PRU t( )2 1 p `~!2( )=T 1PRU t( )2
where RUt( ), t = 2; :::; T , are the OLS residuals from a regression of DUt( )onto 1.
In fact, RUt( )can be written in the simple form
RUt( ) =
1; t T ; t > T
which implies thatPRUt( ) ut= uT + uT + (1 )u1 = Op(1), since ut is I(0).
Moreover, ~!2( )! limp T !1T 1E(PT
t=2 ut)
2 = 0, since u
tis over-di¤erenced when
utis I(0). However, it follows from Leybourne et al. (2004) that `~!2( ) p
! 2P1s=1s 0s
where 0s= E( ut ut s). Consequently, (T =`)1=2t1( ) = Op(1), which establishes the
result in (b).
(ii) In order to establish the result in (a), notice …rst that (`=T )1=2t0( ) = + T 5=2PRTt( )ut T 3PRT t( )2 1 q (`T ) 1!^2( )=T 3PRT t( )2 : (A.2) A simple extension of the results in KPSS (pp.168-169) establishes the result that (`T ) 1!^2( )! !d "2R01H(r; )2drwhere H(r; )is a continuous-time residual from the projection of W (r) onto the space spanned by f1; r; (r )1(r > )g. Consequently, and since all the other stochastic terms appearing in (A.2) are of Op(1) with
non-generate limiting distributions, it follows that (`=T )1=2t0( ) = Op(1), from which the
result in (a) follows.
In order to establish the result in (b), observe that t1( ) = + T 1=2PRUt( )"t T 1PRU t( )2 1 p ~ !2( )=T 1PRU t( )2 d ! ( +!" R1 0 RU (r; )dW (r)dr R1 0 RU (r; )2dr ) 1 q !2 "= R1 0 RU (r; )2dr (A.3)
using standard results, including the fact that ~!2( )! !p 2
". Rearranging (A.3) delivers
the stated result in (b).
Proof of Lemma 1. The proof of Lemma 1 follows from trivial extensions to the results in KPSS (pp.164-169) and, for (i)(b), results in Leybourne et al. (2004). The proof is therefore omitted in the interests of brevity.
Proof of Theorem 2. Consider …rst the proof of the results in (i)(a) and (b)(ii). Here the joint distributions of the sequences of statistics used in forming the statistics t0 and t1 follow from the …xed representations given in Theorem 1, using arguments proved in Zivot and Andrews (1992). The stated results in (i)(a) and (b)(ii) then follow directly from Theorem 1 (i)(a) and (ii)(b), respectively, using applications of the CMT, noting the continuity of the sup function. The result in (i)(b) follows from Theorem 1 (i)(b) and the result that `~!2( ) !p 2P1s=1s 0
s uniformly in : Hence,
t1( ) = Opf(`=T )1=2g uniformly in . Finally, for the result in (ii)(a) we appeal to
Theorem 1 (ii)(a) and the fact that because (`T ) 1!^2( ) d
! !2 " R1 0 H(r; ) 2dr uniformly in . So, t0( ) = Opf(T=`)1=2g uniformly in .
Proof of Theorem 3.
As we are only concerned with establishing the orders in probability of the statistics under H1, for technical expediency we omit the constant and trend regressors from (1)
and the constant regressor from (4). These particular regressors have no e¤ect on any of the orders involved, but just introduce uninformative algebraic complexities.
(i) To establish the result in part (a) we must …rst derive the order of t0( )under the
…xed alternative H1 : 6= 0. To that end, observe …rst that
T 3=2t0( ) = + P DTt( )ut P DTt( )2 1 q ^ !2( )=T 3PT t=1DTt( )2 p ! p !2 u3(1 ) 3
where we have used the results that ^!2( ) ! !p 2u, and that T 3PTt=1DTt( )2 !
(1 )3=3. Now, it is straightforward to establish that for any
2 , we may write T 3=2jt0( )j = s T 3Py2 t ^2( ) T 2 ^ 2( ) ^ !2( );
from which it is clear that the stated result will hold if both ^2(^) ^2( )and ^!2(^) ^
!2( )are asymptotically negligible. Considering the …rst of these, it is straightforward to show that ^2(^) ^2( ) = T 1 f P DTt(^)ytg2 P DTt(^)2 fPDTt( )ytg2 P DTt( )2 (A.4) from which it is easy to demonstrate that the dominant term of the right member of (A.4) is of the form
2T 1 f P DTt(^)DTt( )g2 P DTt(^)2 X DTt( )2 : (A.5)
After some straightforward but lengthy manipulations, the dominant term of (A.5) can be shown to be given by
2(^cT )2
36 (^ 1)
3
(4 ^ 3) (A.6)
where ^c = ^. Our break fraction estimator ^ can be shown to have the same rate of consistency as the minimum sum of squares break fraction estimator of Perron and Zhu (2005). Consequently, from Theorem 3 of Perron and Zhu (2005, p.75) we have that ^
c = Op(T 3=2), and, hence, that (A.6) is Op(T 1). Thus, ^2(^) ^2( ) = Op(T 1).
Turning to the di¤erence between the estimated LR variances, we have that ^ !2(^) !^2( ) = ^2(^) ^2( ) + 2 T 1 X j=1 h(j=`)f^j(^) ^j( )g = Op(T 1) + Op(T 1)O(`) = Op(`T 1)
which follows since ^j(^) ^j( )is uniformly bounded by an Op(T 1)variate. Hence, it
also holds that ^!2(^) !^2( )! 0, and, as a consequence, Tp 3=2
fjt0(^)j jt0( )jg p
! 0, which establishes the result in (a).
In order to establish the result in (b) we again must …rst determine the order of t1( )
under H1. Observe …rst that,
(`T ) 1=2t1( ) = + P DUt( ) ut P DUt( )2 1 p `~!2( )=T 1PDU t( )2 = f + op(1)gOp(1) = Op(1):
For any 2 it can be shown that
(`T ) 1=2jt1( )j = s T 1P y2 t ~2( ) 1 ~2( ) `~!2( )
so that, as in proof of part (a), we must establish the behaviour of the di¤erence between the OLS and LR variance estimators evaluated at and ~. Considering the di¤erence between the OLS variance estimators …rst, we have that
~2(~) ~2( ) = T 1 f P DUt(~) ytg2 P DUt(~)2 fPDUt( ) ytg2 P DUt( )2 ; (A.7)
the dominant term in the right member of which can be shown to be given by
2 ~c( 1)
(2 ~ 1) (A.8)
where ~c = ~. Consequently, ~2(~) ~2( ) = Op(T 1=2) owing to the fact that
~
c = Op(T 1=2), as is straightforward but tedious to establish. As regards the di¤erence
between the LR variance estimators, we have that
`f~!2(~) !~2( )g = `f~2(~) ~2( )g + 2` T 2 X j=1 h(j=`)f~j(~) ~j( )g = `Op(T 1=2) + `Op(T 1=2)O(`) = Op(1)
since ` = O(T1=4). Consequently, (`T ) 1=2
fjt1(~)j jt1( )jg = Op(1), establishing (b).
(ii) In order to prove (a), we again establish …rst the behaviour of t0( ) under H1.
Observing …rst that (`1=2=T )t0( ) = + P DTt( )ut P DTt( )2 1 q (`T ) 1!^2( )=T 3PT t=1DTt( )2 = f + op(1)gOp(1) = Op(1):
Again, for any 2 , we may write (`1=2=T )jt0( )j = s T 3Py2 t T 1^2( ) T 1 T 1^2( ) (`T ) 1!^2( )
so again we need to establish the behaviour of the di¤erence between the OLS and LR variance estimators evaluated at and ~. Using (A.4)-(A.6) we obtain that the dominant term of T 1
f^2(^) ^2( )g in this case is given by
2c^2T
36 (^ 1)
3
(4 ^ 3) :
Next, again utilizing the results from Theorem 3 of Perron and Zhu (2005), we may show that ^c = Op(T 1=2), and, hence, we obtain that T 1f^2(^) ^2( )g = Op(1).
As regards the di¤erence between the LR variance estimators, evaluated at and ^, we have that (`T ) 1f^!2(^) !^2( )g = (`T ) 1f^2(^) ^2( )g + 2(`T ) 1 T 1 X j=1 h(j=`)f^j(^) ^j( )g = Op(` 1) + ` 1Op(1)O(`) = Op(1) so it follows that (`1=2=T ) fjt0(^)j jt0( )jg = Op(1), establishing (a).
Turning …nally to the result in (b), we note …rst that the statistic evaluated at the true break point can be written as
T 1=2t1( ) = + P DUt( )"t P DUt( )2 1 p ~ !2( )=T 1PDU t( )2 p ! (1 ) 1=2 !" : Notice again that for any 2 we may write
T 1=2jt1( )j = s T 1P y2 t ~2( ) 1 ~2( ) ~ !2( )
so again we need to establish the behaviour of the di¤erence between the OLS and LR variance estimators evaluated at and ~. Now from (A.7) and (A.8) it follows that the dominant term of ~2(~) ~2( ) is given by ( 2~c( 1))= (2 ~ 1).
Consequently, ~2(~) ~2( ) = Op(T 1=2)owing to the fact that as in the case of I(0)
data, ~c = Op(T 1=2). Moreover, for the di¤erence between the LR variance estimators,
we have that ~ !2(~) !~2( ) = ~2(~) ~2( ) + 2 T 2 X j=1 h(j=`)f~j(~) ~j( )g = Op(T 1=2) + Op(T 1=2)O(`) = Op(`T 1=2)
so that ~!2(~) !~2( ) ! 0. Hence, Tp 1=2 fjt1(~)j jt1( )jg p ! 0, which establishes the result in (b).
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Table 1. Asymptotic critical values and mξ values for the tλ tests.
Model A Model B ξ Critical value mξ Critical value mξ
0.10 2.284 0.835 2.904 1.062 0.05 2.563 0.853 3.162 1.052 0.01 3.135 0.890 3.654 1.037
Table 2. Empirical sizes of nominal 0.05-level tests.
Panel A. Model A
T = 150 T = 300
c θ tλ MeanWBG0 .02 SupWBG0 .10 SupWBG0 .06 tλ MeanWBG0 .02 SupWBG0 .10 SupWBG0 .06
0 −0.8 0.161 0.035 0.043 0.041 0.114 0.038 0.040 0.039 −0.4 0.155 0.037 0.045 0.042 0.110 0.039 0.041 0.040 0.0 0.139 0.046 0.050 0.050 0.098 0.043 0.045 0.044 0.4 0.094 0.093 0.076 0.084 0.063 0.071 0.062 0.065 0.8 0.133 0.304 0.201 0.251 0.112 0.309 0.184 0.239 10 −0.8 0.030 0.013 0.011 0.008 0.016 0.015 0.011 0.009 −0.4 0.030 0.014 0.011 0.009 0.016 0.016 0.012 0.009 0.0 0.030 0.018 0.013 0.011 0.016 0.018 0.013 0.010 0.4 0.032 0.042 0.025 0.025 0.020 0.033 0.019 0.018 0.8 0.060 0.126 0.066 0.085 0.077 0.188 0.068 0.098 20 −0.8 0.025 0.009 0.008 0.005 0.016 0.010 0.007 0.004 −0.4 0.025 0.009 0.009 0.005 0.016 0.011 0.008 0.005 0.0 0.023 0.014 0.010 0.007 0.017 0.013 0.009 0.005 0.4 0.029 0.035 0.018 0.016 0.023 0.025 0.013 0.011 0.8 0.032 0.048 0.042 0.040 0.075 0.134 0.042 0.056 T −0.8 0.018 0.009 0.025 0.016 0.025 0.017 0.032 0.023 −0.4 0.018 0.012 0.028 0.019 0.026 0.019 0.035 0.026 0.0 0.015 0.018 0.036 0.028 0.022 0.028 0.038 0.034 0.4 0.005 0.014 0.046 0.040 0.009 0.033 0.046 0.043 0.8 0.000 0.000 0.089 0.055 0.000 0.004 0.082 0.073 Panel B. Model B T = 150 T = 300 c θ tλ MeanWBG0 .02 SupW 0 .36 J MeanW 0 .18 BG tλ MeanWBG0 .02 SupW 0 .36 J MeanW 0 .18 BG 0 −0.8 0.161 0.034 0.057 0.039 0.114 0.037 0.055 0.040 −0.4 0.156 0.036 0.056 0.040 0.110 0.039 0.055 0.041 0.0 0.140 0.045 0.053 0.045 0.099 0.044 0.053 0.044 0.4 0.108 0.099 0.047 0.066 0.068 0.076 0.047 0.055 0.8 0.229 0.395 0.045 0.150 0.230 0.429 0.043 0.136 10 −0.8 0.032 0.012 0.029 0.018 0.018 0.016 0.029 0.019 −0.4 0.032 0.013 0.028 0.019 0.018 0.016 0.029 0.019 0.0 0.035 0.020 0.027 0.021 0.021 0.020 0.028 0.020 0.4 0.050 0.056 0.025 0.032 0.034 0.043 0.027 0.026 0.8 0.128 0.170 0.029 0.064 0.213 0.308 0.029 0.062 20 −0.8 0.028 0.009 0.024 0.016 0.020 0.008 0.027 0.015 −0.4 0.030 0.010 0.023 0.017 0.020 0.009 0.027 0.015 0.0 0.035 0.013 0.023 0.020 0.026 0.012 0.026 0.017 0.4 0.060 0.041 0.022 0.031 0.049 0.034 0.025 0.023 0.8 0.067 0.050 0.024 0.048 0.197 0.198 0.026 0.049 T −0.8 0.030 0.005 0.024 0.034 0.039 0.012 0.031 0.038 −0.4 0.032 0.007 0.024 0.036 0.042 0.015 0.032 0.040 0.0 0.032 0.013 0.026 0.042 0.042 0.024 0.032 0.043 0.4 0.014 0.010 0.028 0.047 0.017 0.030 0.039 0.046 0.8 0.001 0.000 0.021 0.033 0.000 0.003 0.044 0.036
T ab le 3. App lication of tes ts to US m acro ec on om ic time se rie s. Mo del A M o del B Seri e s ξ tλ Me a n W 0 .02 BG Sup W 0 .10 BG Sup W 0 .06 BG tλ Me an W 0 .02 BG Sup W 0 .36 J Me a n W 0 .18 BG Real ou tp ut 0.05 1.085 0. 024 0.113 0. 076 1. 350 0.082 671.358 0. 581 0.01 1.130 0.010 0. 050 0.033 1.334 0.029 513. 746 0.310 (-) (-) Un e mplo y m en t 0.05 2.417 0. 516 2.310 1. 598 2. 967 0.600 2153.949 6. 227 0.01 2.522 0.217 1. 053 0.715 2.925 0.224 1526. 300 3 .463 (-) (-) Mon e y sup pl y 0.05 5.710 ∗∗ 0.006 0.719 0.246 7.047 ∗∗ 0.003 14410.803 ∗∗ 0.752 0.01 5.957 ∗∗ ∗ 0.000 0.068 0.022 6.947 ∗∗ ∗ 0.000 163.638 0.12 1 (19 86M12 ) (1 986M1 2) In teres t rate 0.05 2.884 ∗∗ 0.339 1.981 1.100 3.665 ∗∗ 0.264 473.080 2.39 9 0.01 3.009 0.113 0. 731 0.396 3.612 0.072 268. 033 1.110 (20 00M7) (2 000M7 ) Comm o d it y pr ice s 0.05 2.603 ∗∗ 1.894 10.543 6.471 3.215 ∗∗ 1.368 1185.617 10.04 1 0.01 2.716 0. 808 4. 864 2 .932 3. 169 0.478 742. 748 5. 371 (19 74M2) (1 974M2 ) Consumer pri c es 0.05 7.235 ∗∗ 0.332 9.717 4.501 8.925 ∗∗ 0.158 12890.790 ∗∗ 4.313 0.01 7.549 ∗∗ ∗ 0.064 2.169 0.970 8.798 ∗∗ ∗ 0.020 4344.766 1.2 43 (19 82M7) (1 982M6 ) E arni ngs 0.05 9.536 ∗∗ 0.024 3.721 1.534 11.284 ∗∗ 0.637 2657.756 12.1 01 0.01 9.950 ∗∗ ∗ 0.002 0.423 0.166 11.123 ∗∗ ∗ 0.103 1532.787 4.0 86 (19 82M1) (1981 M12) Notes : ∗∗ and ∗∗ ∗ den ote s re je ction at th e 0.05-lev el an d 0.01-lev e l re sp ectiv ely . Wh e re rejections are ob tain e d for the tλ tes t, th e es ti m ated b reak da te is re p orted in paren th e se s.
(a) T = 150, τ∗ = 0.25 (b) T = 300, τ∗= 0.25
(c) T = 150, τ∗ = 0.5 (d) T = 300, τ∗= 0.5
(e) T = 150, τ∗ = 0.75 (f) T = 300, τ∗ = 0.75
(a) T = 150, τ∗ = 0.25 (b) T = 300, τ∗= 0.25
(c) T = 150, τ∗ = 0.5 (d) T = 300, τ∗= 0.5
(e) T = 150, τ∗ = 0.75 (f) T = 300, τ∗ = 0.75
(a) T = 150, τ∗ = 0.25 (b) T = 300, τ∗= 0.25
(c) T = 150, τ∗ = 0.5 (d) T = 300, τ∗= 0.5
(e) T = 150, τ∗ = 0.75 (f) T = 300, τ∗ = 0.75
(a) T = 150, τ∗ = 0.25 (b) T = 300, τ∗= 0.25
(c) T = 150, τ∗ = 0.5 (d) T = 300, τ∗= 0.5
(e) T = 150, τ∗ = 0.75 (f) T = 300, τ∗ = 0.75
Figure 4. Size-adjusted power: Model A, c = 0, tλ: , MeanWBG0 .02: – – , SupWBG0 .10: - - - ,
(a) T = 150, τ∗ = 0.25 (b) T = 300, τ∗= 0.25
(c) T = 150, τ∗ = 0.5 (d) T = 300, τ∗= 0.5
(e) T = 150, τ∗ = 0.75 (f) T = 300, τ∗ = 0.75
Figure 5. Power: Model B, c = 0, tλ: , MeanWBG0 .02: – – , SupWJ0 .36: - - - ,
(a) T = 150, τ∗ = 0.25 (b) T = 300, τ∗= 0.25
(c) T = 150, τ∗ = 0.5 (d) T = 300, τ∗= 0.5
(e) T = 150, τ∗ = 0.75 (f) T = 300, τ∗ = 0.75
Figure 6. Power: Model B, c = 10, tλ: , MeanWBG0 .02: – – , SupWJ0 .36: - - - ,
(a) T = 150, τ∗ = 0.25 (b) T = 300, τ∗= 0.25
(c) T = 150, τ∗ = 0.5 (d) T = 300, τ∗= 0.5
(e) T = 150, τ∗ = 0.75 (f) T = 300, τ∗ = 0.75
Figure 7. Power: Model B, c = 20, tλ: , MeanWBG0 .02: – – , SupWJ0 .36: - - - ,
(a) T = 150, τ∗ = 0.25 (b) T = 300, τ∗= 0.25
(c) T = 150, τ∗ = 0.5 (d) T = 300, τ∗= 0.5
(e) T = 150, τ∗ = 0.75 (f) T = 300, τ∗ = 0.75
Figure 8. Size-adjusted power: Model B, c = 0, tλ: , MeanWBG0 .02: – – , SupWJ0 .36: - - - ,
(a) Real output (b) Unemployment
(c) Money supply (d) Interest rate
(e) Commodity prices (f) Consumer prices
(g) Earnings
